Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

Find the functions $ f:\mathbb{Z}\longrightarrow\mathbb{Z}_{\ge 0} $ that satisfy the following two conditions: $ \text{(a)} f(m+n)=f(n)+f(m)+2mn,\quad\forall m,n\in\mathbb{Z} $ $ \text{(b)} f(f(1))-f(1) $ is a perfect square [i]Marin Ionescu[/i]

Compute the smallest positive integer $n$ for which there are at least two odd primes $p$ such that $\sum_{k=1}^{n} (-1)^{v_p(k!)} < 0$. Note: for a prime $p$ and a positive integer $m$, $v_p(m)$ is the exponent of the largest power of $p$ that divides $m$; for example, $v_3(18) = 2$.

To celebrate the $20$th LMT, the LHSMath Team bakes a cake. Each of the $n$ bakers places $20$ candles on the cake. When they count, they realize that there are $(n -1)!$ total candles on the cake. Find $n$. [i]Proposed by Christopher Cheng[/i]

Let $n > 1$ be a positive integer and $p$ a prime number such that $n | (p - 1) $and $p | (n^6 - 1)$. Prove that at least one of the numbers $p- n$ and $p + n$ is a perfect square.

In the country of East Westmore, statisticians estimate there is a baby born every 8 hours and a death every day. To the nearest hundred, how many people are added to the population of East Westmore each year? $\textbf{(A)}\hspace{.05in}600 \qquad \textbf{(B)}\hspace{.05in}700 \qquad \textbf{(C)}\hspace{.05in}800 \qquad \textbf{(D)}\hspace{.05in}900 \qquad \textbf{(E)}\hspace{.05in}1000 $

Janet rolls a standard 6-sided die 4 times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal 3? $\textbf{(A) }\frac{2}{9}\qquad\textbf{(B) }\frac{49}{216}\qquad\textbf{(C) }\frac{25}{108}\qquad\textbf{(D) }\frac{17}{72}\qquad\textbf{(E) }\frac{13}{54}$

Let $O$ be the circumcenter of a triangle $ABC$. The projections of points $D$ and $X$ to the sidelines of the triangle lie on lines $\ell $ and $L $ such that $\ell // XO$. Prove that the angles formed by $L$ and by the diagonals of quadrilateral $ABCD$ are equal.

Given a triangle $ABC$ isosceles at $A.$ A point $P$ lying inside the triangle such that $\angle PBC=\angle PCA$ and let $M$ be the midpoint of $BC.$ Prove that: $\angle APB+ \angle MPC =180^{\circ}.$

For any positive integer $n$, let $p(n)$ be the product of its digits in base-$10$ representation. Find the maximum possible value of $\frac{p(n)}{n}$ over all integers $n \ge 10$.

Determine the smallest prime $p$ such that $2018!$ is divisible by $p^{3}$ , but not divisible by $p^{4}$.

Let $ABCD$ be a cyclic quadrilateral an let $P$ be a point on the side $AB.$ The diagonals $AC$ meets the segments $DP$ at $Q.$ The line through $P$ parallel to $CD$ mmets the extension of the side $CB$ beyond $B$ at $K.$ The line through $Q$ parallel to $BD$ meets the extension of the side $CB$ beyond $B$ at $L.$ Prove that the circumcircles of the triangles $BKP$ and $CLQ$ are tangent .

Determine all ordered triples $(a, b, c)$ of real numbers such that whenever a function $f : \mathbb{R} \to \mathbb{R}$ satisfies $$|f(x) - f(y)| \le a(x - y)^2 + b(x - y) + c$$ for all real numbers $x$ and $y$, then $f$ must be a constant function.

There exists a fraction $x$ that satisfies $ \sqrt{x^2+5} - x = \tfrac{1}{3}$. What is the sum of the numerator and denominator of this fraction? $\textbf{(A) }8\qquad\textbf{(B) }21\qquad\textbf{(C) }25\qquad\textbf{(D) }32\qquad\textbf{(E) }34$

Given a scalene triangle $\vartriangle ABC$ with orthocenter $H$. The midpoint of $BC$ is denoted by $M$. $AH$ intersects the circumcircle at $D \ne A$ and $DM$ intersects circumcircle of $\vartriangle ABC$ at $T\ne D$. Now, assume the reflection points of $M$ with respect to $AB,AC,AH$ are $F,E,S$. Show that the midpoints of $BE,CF,AM,TS$ are concyclic. [img]https://3.bp.blogspot.com/-v7D_A66nlD0/XnYNJussW9I/AAAAAAAALeQ/q6DMQ7w6QtI5vLwBcKqp4010c3XTCj3BgCK4BGAYYCw/s1600/imoc2019g2.png[/img]

Given is a prime number $p$. Find the number of positive integer solutions $(a, b, c, d)$ of the system of equations $ac+bd = p(a+c)$ and $bc-ad = p(b-d)$.

Let $\Gamma$ be a simple curve, lying inside a circle of radius $r$, rectifiable and of length $\ell$. Prove that if $\ell > kr\pi$, then there exists a circle of radius $r$ which intersects $\Gamma$ in at least $k+1$ distinct points.

If positive integers, $a,b,c$ are pair-wise co-prime, and, \[\ a^2|(b^3+c^3), b^2|(a^3+c^3), c^2|(a^3+b^3) \] find $a,b,$ and $c$

In the plane are given two mutually perpendicular lines and $n$ rectangles with sides parallel to the two lines. Show that if every two rectangles have a common point, then all the rectangles have a common point.

An array $ n\times n$ is given, consisting of $ n^2$ unit squares. A [i]pawn[/i] is placed arbitrarily on a unit square. The pawn can move from a square of the $ k$-th column to any square of the $ k$-th row. Show that there exists a sequence of $ n^2$ moves of the pawn so that all the unit squares of the array are visited once, the pawn returning to its original position. [b]Dinu Serbanescu[/b]

Let $ f: \mathbb R \to \mathbb R$ be a function, two times derivable on $ \mathbb R$ for which there exist $ c\in\mathbb R$ such that \[ \frac { f(b)\minus{}f(a) }{b\minus{}a} \neq f'(c) ,\] for all $ a\neq b \in \mathbb R$. Prove that $ f''(c)\equal{}0$.

Let $a,b,c$ be distinct real numbers such that $a+b+c>0$. Let $M$ be the set of $3\times 3$ matrices with the property that each line and each column contain all given numbers $a,b,c$. Find $\{\max \{ \det A \mid A \in M \}$ and the number of matrices which realise the maximum value. [i]Mircea Becheanu[/i]

1) proove for positive $a, b, c, d$ $ \frac{a}{b+c} + \frac{b}{c+d} + \frac{c}{d+a} + \frac{d}{a+b} \ge 2$

An integer is square-free if it is not divisible by $a^2$ for any integer $a>1$. Let $S$ be the set of positive square-free integers. Determine, with justification, the value of\[\sum_{k\epsilon S}\left[\sqrt{\frac{10^{10}}{k}}\right]\]where $[x]$ denote the greatest integer less than or equal to $x$

Consider the graphs of (1): $ y\equal{}x^2\minus{}\frac{1}{2}x\plus{}2$ and (2) $ y\equal{}x^2\plus{}\frac{1}{2}x\plus{}2$ on the same set of axis. These parabolas are exactly the same shape. Then: $ \textbf{(A)}\ \text{the graphs coincide.} \\ \textbf{(B)}\ \text{the graph of (1) is lower than the graph of (2).} \\ \textbf{(C)}\ \text{the graph of (1) is to the left of the graph of (2).} \\ \textbf{(D)}\ \text{the graph of (1) is to the right of the graph of (2).} \\ \textbf{(E)}\ \text{the graph of (1) is higher than the graph of (2).}$